On knot Floer homology and cabling II

TL;DR

This paper investigates how knot Floer homology invariants behave under cabling operations, revealing isomorphisms in filtered subcomplexes for large parameters and exploring implications for concordance and geometric properties.

Contribution

It proves isomorphisms of filtered subcomplexes in knot Floer homology for large cabling parameters and analyzes the impact on the Ozsvath-Szabo concordance invariant.

Findings

Filtered subcomplexes are isomorphic for large |n| in (p,pn+1) cable knots.

The Ozsvath-Szabo concordance invariant's behavior under cabling is characterized.

Applications include insights into quasipositivity, complex curves, and L-space surgeries.

Abstract

We continue our study of the knot Floer homology invariants of cable knots. For large |n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p,pn+1) cable of a knot, K, are isomorphic to those of K. This result allows us to obtain information about the behavior of the Ozsvath-Szabo concordance invariant under cabling, which has geometric consequences for the cabling operation. Applications considered include quasipositivity in the braid group, the knot theory of complex curves, smooth concordance, and lens space (or, more generally, L-space) surgeries.